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Center of mass and constant mean curvature foliations for isolated systems.

机译：隔离系统的质心和恒定平均曲率叶面。

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摘要

We discuss center of mass for asymptotically flat manifolds satisfying the Regge-Teitelboim condition from two different points of view: the Hamiltonian formulation as flux integrals at infinity, and the geometric description using foliations by surfaces with constant mean curvature. The equivalence of those different notions of center of mass is also proven. Physicists have proposed a notion of center of mass as a flux integral at infinity from the centroid for the distribution of energy. We propose another flux integral for center of mass involving the three dimensional Einstein tensor. This notion is more intrinsic because it has a coordinate-free expression and natural properties. Moreover, it is equivalent to the previous one. The main tool is a new density theorem for data satisfying the Regge-Teitelboim condition.;The new density theorem says that the solutions with harmonic asymptotics to the constraint equations are dense among solutions which satisfies the Regge-Teitelboim condition in some weighted Sobolev spaces. Moreover, mass, linear momentum, center of mass, and angular momentum converge to the ones from the original initial data.;Using this density theorem as one ingredient, we relate some integrals involving mean curvature of Euclidean spheres to center of mass. Then we prove the existence and uniqueness of foliations by surfaces with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition. We also show that the foliation is asymptotically concentric, and its geometric center is equal to the other two notions of center of mass. In addition, the unique constant mean curvature foliation may provide a polar coordinate system for such manifolds, and potentially enable us to understand their geometric structure.
机译：我们从两个不同的角度讨论满足Regge-Teitelboim条件的渐近平流形的质心：两种形式的汉密尔顿公式（作为无穷大处的通量积分），以及使用平均曲率恒定的面的叶面进行的几何描述。还证明了这些不同质心概念的等效性。物理学家提出了质心的概念，即从质心起无穷远处的通量积分，用于能量分配。我们提出了另一个涉及三维爱因斯坦张量的重心通量积分。这个概念更内在，因为它具有无坐标的表达方式和自然属性。而且，它等效于前一个。主要工具是用于满足Regge-Teitelboim条件的数据的新密​​度定理。;新的密度定理说，在某些加权Sobolev空间中，满足Regge-Teitelboim条件的解决方案中具有约束的调和渐近渐近解。而且，质量，线性动量，质心和角动量收敛于原始初始数据。；使用该密度定理作为一种成分，我们将一些涉及欧几里德球体平均曲率的积分与质心联系起来。然后我们证明了满足Regge-Teitelboim条件的渐近平坦流形具有平均曲率恒定的表面的叶面的存在和唯一性。我们还表明，叶面渐近同心，并且其几何中心等于质心的其他两个概念。此外，独特的恒定平均曲率叶面可以为此类歧管提供极坐标系统，并有可能使我们了解其几何结构。

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  • 作者

    Huang, Lan-Hsuan.;

  • 作者单位

    Stanford University.;

  • 授予单位 Stanford University.;
  • 学科 Mathematics.;Physics Theory.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 95 p.
  • 总页数 95
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

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